Thread: Create a 3x3 grid of unique integers squared, each row, column, diag. must equal N

1. Create a 3x3 grid of unique integers squared, each row, column, diag. must equal N

Not sure if this is the correct category, if its not sorry.

If you create a 3x3 grid of unique integers squared, each row, column and diagonal has to be equal to N.
What is one solution to this problem and what is a general solution to this problem?

 n a2 b2 c2 n d2 e2 f2 n g2 h2 i2 n n n n n

Using systems of equtions I was able to find that e2= 1/3n.

a2 +b2 +c2 =n
-(a2 +e2 +i2 =n)
-(b2 +e2 +h2 =n)
-(c2 +e2 +g2 =n)
g2 +h2 +i2 =n
=
-3e2 =n

After this I have been able to write everything in terms of a, c and n but don't know where to go next.

 n a2 b2 c2 n d2 1n/3 f2 n g2 h2 i2 n n n n n

2. First thing to notice is that you have 9 unknowns, but only 8 equations. So you won't, in general, have a single unique solution. You'll probably have infinitely-many solutions.

What you did is fine. However, each equation can be used to reduce the number of variables, one at a time.

a^2 + b^2 + c^2 = n

a^2 = n - (b^2 + c^2)

In principle you could just substitute this in everywhere that you see a^2, and that variable will have been eliminated. Repeat.

3. Originally Posted by j-astron
In principle you could just substitute this in everywhere that you see a^2, and that variable will have been eliminated. Repeat.
I have substituted and got every thing in terms of a and c.

Im in school right now but when I get home I'll share what I got.

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